Geodesic
Potentially diagonalisable lifts with controlled Hodge-Tate weights
Documenta mathematica, Tome 26 (2021), pp. 795-827.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Motivated by the weight part of Serre's conjecture we consider the following question. Let $K/\mathbb{Q}_p$ be a finite extension and suppose $\overline{\rho} : G_K \rightarrow \operatorname{GL}_n(\overline{\mathbb{F}}_p)$ admits a crystalline lift with Hodge-Tate weights contained in the range $[0,p]$. Does $\overline{\rho}$ admit a potentially diagonalisable crystalline lift of the same Hodge-Tate weights? We answer this question in the affirmative when $K = \mathbb{Q}_p$ and $n \leq 5$, and $\overline{\rho}$ satisfies a mild 'cyclotomic-free' condition. We also prove partial results when $K/\mathbb{Q}_p$ is unramified and $n$ is arbitrary.
Classification : 11F80, 11F33
Keywords: congruences between crystalline representations, integral \(p\)-adic Hodge theory, breuil-kisin modules 
@article{DOCMA_2021__26__a34,
     author = {Bartlett, Robin},
     title = {Potentially diagonalisable lifts with controlled {Hodge-Tate} weights},
     journal = {Documenta mathematica},
     pages = {795--827},
     publisher = {mathdoc},
     volume = {26},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2021__26__a34/}
}
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Bartlett, Robin. Potentially diagonalisable lifts with controlled Hodge-Tate weights. Documenta mathematica, Tome 26 (2021), pp. 795-827. http://geodesic.mathdoc.fr/item/DOCMA_2021__26__a34/